Chapter 40 De Moivres Theorem simple applications 1122020

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Chapter 40 De Moivre’s Theorem & simple applications 11/2/2020 By Chtan FYHS-Kulai 1

Chapter 40 De Moivre’s Theorem & simple applications 11/2/2020 By Chtan FYHS-Kulai 1

In mathematics, de Moivre‘s formula, named after Abraham de Moivre. 11/2/2020 By Chtan FYHS-Kulai

In mathematics, de Moivre‘s formula, named after Abraham de Moivre. 11/2/2020 By Chtan FYHS-Kulai 2

The formula is important because it connects complex numbers and trigonometry. The expression "cos

The formula is important because it connects complex numbers and trigonometry. The expression "cos x + i sin x" is sometimes abbreviated to "cis x". 11/2/2020 By Chtan FYHS-Kulai 3

By expanding the left hand side and then comparing the real and imaginary parts

By expanding the left hand side and then comparing the real and imaginary parts under the assumption that x is real, it is possible to derive useful expressions for cos(nx) and sin(nx) in terms of cos(x) and sin(x). 11/2/2020 By Chtan FYHS-Kulai 4

Furthermore, one can use a generalization of this formula to find explicit expressions for

Furthermore, one can use a generalization of this formula to find explicit expressions for the n-th roots of unity, that is, complex numbers z such that zn = 1. 11/2/2020 By Chtan FYHS-Kulai 5

De Moivre’s theorem For all values of n, the value, or one of the

De Moivre’s theorem For all values of n, the value, or one of the values in the case where n is fractional, of is 11/2/2020 By Chtan FYHS-Kulai 6

Proofing of De Moivre’s Theorem 11/2/2020 By Chtan FYHS-Kulai 7

Proofing of De Moivre’s Theorem 11/2/2020 By Chtan FYHS-Kulai 7

Now, let us prove this important theorem in 3 parts. 1. When n is

Now, let us prove this important theorem in 3 parts. 1. When n is a positive integer 2. When n is a negative integer 3. When n is a fraction 11/2/2020 By Chtan FYHS-Kulai 8

Case 1 : if n is a positive integer 11/2/2020 By Chtan FYHS-Kulai 9

Case 1 : if n is a positive integer 11/2/2020 By Chtan FYHS-Kulai 9

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Continuing this process, when n is a positive integer, 11/2/2020 By Chtan FYHS-Kulai 12

Continuing this process, when n is a positive integer, 11/2/2020 By Chtan FYHS-Kulai 12

Case 2 : if n is a negative integer Let n=-m where m is

Case 2 : if n is a negative integer Let n=-m where m is positive integer 11/2/2020 By Chtan FYHS-Kulai 13

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Case 3 : if n is a fraction equal to p/q, p and q

Case 3 : if n is a fraction equal to p/q, p and q are integers 11/2/2020 By Chtan FYHS-Kulai 15

Raising the RHS to power q we have, but, 11/2/2020 By Chtan FYHS-Kulai 16

Raising the RHS to power q we have, but, 11/2/2020 By Chtan FYHS-Kulai 16

Hence, De Moivre’s Theorem applies when n is a rational fraction. 11/2/2020 By Chtan

Hence, De Moivre’s Theorem applies when n is a rational fraction. 11/2/2020 By Chtan FYHS-Kulai 17

Proofing by mathematic al induction 11/2/2020 By Chtan FYHS-Kulai 18

Proofing by mathematic al induction 11/2/2020 By Chtan FYHS-Kulai 18

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The hypothesis of Mathematical Induction has been satisfied , and we can conclude that

The hypothesis of Mathematical Induction has been satisfied , and we can conclude that 11/2/2020 By Chtan FYHS-Kulai 21

e. g. 1 Let z = 1 − i. Find . Soln: First write

e. g. 1 Let z = 1 − i. Find . Soln: First write z in polar form. 11/2/2020 By Chtan FYHS-Kulai 22

Polar form : Applying de Moivre’s Theorem gives : 11/2/2020 By Chtan FYHS-Kulai 23

Polar form : Applying de Moivre’s Theorem gives : 11/2/2020 By Chtan FYHS-Kulai 23

It can be verified directly that 11/2/2020 By Chtan FYHS-Kulai 24

It can be verified directly that 11/2/2020 By Chtan FYHS-Kulai 24

Properties of 11/2/2020 By Chtan FYHS-Kulai 25

Properties of 11/2/2020 By Chtan FYHS-Kulai 25

If then 11/2/2020 By Chtan FYHS-Kulai 26

If then 11/2/2020 By Chtan FYHS-Kulai 26

Hence, 11/2/2020 By Chtan FYHS-Kulai 27

Hence, 11/2/2020 By Chtan FYHS-Kulai 27

Similarly, if Hence, 11/2/2020 By Chtan FYHS-Kulai 28

Similarly, if Hence, 11/2/2020 By Chtan FYHS-Kulai 28

We have, Maximum value of cosθ is 1, minimum value is -1. Hence, normally

We have, Maximum value of cosθ is 1, minimum value is -1. Hence, normally 11/2/2020 By Chtan FYHS-Kulai 29

What happen, if the value of is more than 2 or less than -2

What happen, if the value of is more than 2 or less than -2 ? 11/2/2020 By Chtan FYHS-Kulai 30

e. g. 2 Given that Prove that 11/2/2020 By Chtan FYHS-Kulai 31

e. g. 2 Given that Prove that 11/2/2020 By Chtan FYHS-Kulai 31

e. g. 3 If 11/2/2020 , find By Chtan FYHS-Kulai 32

e. g. 3 If 11/2/2020 , find By Chtan FYHS-Kulai 32

Do take note of the following : 11/2/2020 By Chtan FYHS-Kulai 33

Do take note of the following : 11/2/2020 By Chtan FYHS-Kulai 33

e. g. 4 11/2/2020 By Chtan FYHS-Kulai 34

e. g. 4 11/2/2020 By Chtan FYHS-Kulai 34

Applications of De Moivre’s theorem 11/2/2020 By Chtan FYHS-Kulai 35

Applications of De Moivre’s theorem 11/2/2020 By Chtan FYHS-Kulai 35

We will consider three applications of De Moivre’s Theorem in this chapter. 1. Expansion

We will consider three applications of De Moivre’s Theorem in this chapter. 1. Expansion of 2. Values of . 3. Expressions for terms of multiple angles. 11/2/2020 . By Chtan FYHS-Kulai in 36

Certain trig identities can be derived using De Moivre’s theorem. In particular, expression such

Certain trig identities can be derived using De Moivre’s theorem. In particular, expression such as can be expressed in terms of : 11/2/2020 By Chtan FYHS-Kulai 37

e. g. 5 Use De Moivre’s Thorem to find an identity for in terms

e. g. 5 Use De Moivre’s Thorem to find an identity for in terms of 11/2/2020 . By Chtan FYHS-Kulai 38

e. g. 6 Soln: Find all complex cube roots of 27 i. We are

e. g. 6 Soln: Find all complex cube roots of 27 i. We are looking for complex number z with the property Strategy : First we write 27 i in polar form : - 11/2/2020 By Chtan FYHS-Kulai 39

Now suppose Satisfies. Then, by De Moivre’s Theorem, 11/2/2020 By Chtan FYHS-Kulai 40

Now suppose Satisfies. Then, by De Moivre’s Theorem, 11/2/2020 By Chtan FYHS-Kulai 40

This means : where k is an integer. Possibilities are : k=0, k=1, k=2

This means : where k is an integer. Possibilities are : k=0, k=1, k=2 11/2/2020 By Chtan FYHS-Kulai 41

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In general : to find the complex nth roots of a non-zero complex number

In general : to find the complex nth roots of a non-zero complex number z. 1. Write z in polar form : 11/2/2020 By Chtan FYHS-Kulai 44

2. z will have n different nth roots (i. e. 3 cube roots, 4

2. z will have n different nth roots (i. e. 3 cube roots, 4 fourth roots, etc. ) 3. All these roots will have the same modulus the positive real nth roots of r). 4. They will have different arguments : 11/2/2020 By Chtan FYHS-Kulai 45

5. The complex nth roots of z are given (in polar form) by …etc

5. The complex nth roots of z are given (in polar form) by …etc 11/2/2020 By Chtan FYHS-Kulai 46

e. g. 7 Find all the complex fourth roots of -16. Soln: Modulus =

e. g. 7 Find all the complex fourth roots of -16. Soln: Modulus = 16 Argument = ∏ 11/2/2020 By Chtan FYHS-Kulai 47

Fourth roots of 16 all have modulus : and possibilities for the arguments are

Fourth roots of 16 all have modulus : and possibilities for the arguments are : 11/2/2020 By Chtan FYHS-Kulai 48

Hence, fourth roots of -16 are : 11/2/2020 By Chtan FYHS-Kulai 49

Hence, fourth roots of -16 are : 11/2/2020 By Chtan FYHS-Kulai 49

e. g. 8 Given that and find the value of m. 11/2/2020 By Chtan

e. g. 8 Given that and find the value of m. 11/2/2020 By Chtan FYHS-Kulai 50

e. g. 9 Solve that 11/2/2020 , hence prove By Chtan FYHS-Kulai 51

e. g. 9 Solve that 11/2/2020 , hence prove By Chtan FYHS-Kulai 51

e. g. 10 Find the cube roots of -1. show that they can be

e. g. 10 Find the cube roots of -1. show that they can be denoted by and prove that 11/2/2020 By Chtan FYHS-Kulai 52

e. g. 11 Solve the following equations, giving any complex roots in the form

e. g. 11 Solve the following equations, giving any complex roots in the form 11/2/2020 By Chtan FYHS-Kulai 53

e. g. 12 Prove that Hence find 11/2/2020 By Chtan FYHS-Kulai 54

e. g. 12 Prove that Hence find 11/2/2020 By Chtan FYHS-Kulai 54

e. g. 13 Show that Use your result to solve the equation 11/2/2020 By

e. g. 13 Show that Use your result to solve the equation 11/2/2020 By Chtan FYHS-Kulai 55

e. g. 14 Use De Moivre’s Theorem to find 11/2/2020 By Chtan FYHS-Kulai 56

e. g. 14 Use De Moivre’s Theorem to find 11/2/2020 By Chtan FYHS-Kulai 56

e. g. 15 11/2/2020 By Chtan FYHS-Kulai 57

e. g. 15 11/2/2020 By Chtan FYHS-Kulai 57

e. g. 16 11/2/2020 By Chtan FYHS-Kulai 58

e. g. 16 11/2/2020 By Chtan FYHS-Kulai 58

e. g. 17 11/2/2020 By Chtan FYHS-Kulai 59

e. g. 17 11/2/2020 By Chtan FYHS-Kulai 59

e. g. 18 11/2/2020 By Chtan FYHS-Kulai 60

e. g. 18 11/2/2020 By Chtan FYHS-Kulai 60

e. g. 19 Express in terms of multiple angles and hence evaluate 11/2/2020 By

e. g. 19 Express in terms of multiple angles and hence evaluate 11/2/2020 By Chtan FYHS-Kulai 61

e. g. 20 Express in terms of and hence evaluate terms of. 11/2/2020 By

e. g. 20 Express in terms of and hence evaluate terms of. 11/2/2020 By Chtan FYHS-Kulai in 62

The end 11/2/2020 By Chtan FYHS-Kulai 63

The end 11/2/2020 By Chtan FYHS-Kulai 63

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